Alexei,
Landau just expresses the frustration that non-mathematicians have when
confronted with mathematical derivations and proofs that strive for rigor.
They may need the results and material but, darn it, it's hard. When they
finish reading a proof they might say: "Well, I suppose so." (Somehow I
can't imagine Landau saying that, but maybe he was speaking for others as
well, or perhaps he was truly frustrated with the presentations."
Let's assume that the mathematical proofs and derivations to be presented
actually are useful with regard to the material. Certain techniques, some
requiring Mathematica, can be a great help.
Background Knowledge: Does the student know what the objects are that the
proof is dealing with? It's one thing to say "Let n be an integer",
something else to say "Let f be a p-linear symmetric mapping." Can we see
some examples? It is one thing to see the derivation of a surprising
relation among objects we thought we knew all about, another thing to end up
with a hazy relation about hazy objects that we didn't know that much about
to begin with. Mathematica can help because we can generate many examples
that should give familiarity with the objects.
Motivation: To really understand most derivations and proofs takes a fair
amount of time. Is the student convinced ahead of time that the result will
be worth the effort? Can he see the problem that the derivation is to solve?
If this is a generalization from an easier problem can he see why one would
want to generalize and why it is not so obvious how to do it?
Known Material and Axioms: Does the reader have a clear knowledge of the
axioms and theorems that go into the derivation? Does he have a clear
distinction between the starting point and what is to be derived? Is he
given references to equations in earlier chapters where the matter is stated
in a different notation and context? And perhaps hard to find - like the
only numbered equation in 25 pages. Does he have to play the role of a real
estate lawyer doing title searches? With Mathematica notebooks space is
inexpensive and all the required material COULD be gathered together perhaps
in a separate window and perhaps in an active form.
Coherent Unit: Is a derivation or proof presented as a coherent unit, or
does it spread out over many pages with incoherent typography? Mathematica
allows a derivation to be presented in outline form and space to be reused
such that it is easy to switch between sections of a derivation or to
display several sections together in adjacent windows. It is possible to
bring the material to the reader when and where they need it.
Active Calculation: Many derivations and proofs in Mathematica can be done
with active calculation using routines and rules provided. People understand
actions much better than successions of static expressions. Carrying out a
derivation actively, as a series of calculated steps, is much more
convincing. Providing the routines for doing this may be a great deal of
work, but they also provide a base for further work in the applied area. The
student studying the proof not only gets the proof but also useful tools.
Rather than a forced purchase she's gotten an extra bonus. Active
calculation also carries out text-proofing, eliminating silly errors and
misprints and producing a document of higher integrity.
Proofs for Mathematicians vs. Proofs for Users: Proofs for mathematicians
carry the burden of correctness and rigor. They can assume quite a bit of
advance knowledge from fellow mathematicians. Proofs for users have an EXTRA
burden of motivation, elaboration and turning the proof into an active
useful tool. If the reader isn't going to use it, then why ask him to read
it in the first place?
Proof Presentation Research: The active and dynamic features of Mathematica
provide means to present proofs and derivations that are not only correct
but easier for readers to follow and use. It is not at all obvious how to do
this. It is very new. Some of the techniques are things I mentioned above.
One good technique with complicated diagrams is to use Checkboxes to turn
various features of the diagram on or off as they pertain to elements of the
proof. There are probably many other techniques that would arise from or be
adapted to special elements in a proof. There is a matter of judgment and
taste - no sense in elaborating the obvious, but what are the sticking
points that need elaboration?
There is a video kindly done for me by Roger Williams illustrating some of
these proof presentation techniques. There are two 8 minute videos but it is
the second that shows the techniques. I've posted these before so some of
you may have already seen them.
http://www.youtube.com/watch?v=-b0B5hp0hAQhttp://www.youtube.com/watch?v=Pm6yrevYcjQ
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/index.html
From: Alexei Boulbitch [mailto:Alexei.Boulbitch at iee.lu]
Dear Community, dear Jon, dear Craig,
After my recent post where I mentioned the letter of L.D. Landau concerning
mathematical education for physicists I received several requests for the
full text. The full (or whatever) text I read years ago in the book of Maiia
Bessarab<http://www.amazon.de/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&field-au
thor=Maiia%20Bessarab&search-alias=books-de> "Lev Landau: Novel-biography"
which can be found here:
http://www.amazon.de/Lev-Landau-Roman-biografiia-Novel-biography/dp/B001MIKD
24/ref=sr_1_2?s=books&ie=UTF8&qid=1344239945&sr=1-2
and in few other places on the web. One can also find sites from where it
can be downloaded for free. It is, however, in Russian. Because of your
requests, and since Landau's ideas today are as valid as they have been 80
years ago, I translated the letter into English for those of you who are
interested to read it, cannot do it in Russian though. Please find it below.
I tried to keep the translation as close to Landau original style and
expressions as I could. Sometimes this lead to phrases that in English sound
cumbersome, some sentences look too long, and special expressions are used,
such as "exorcise" and alike. Landau liked strong (often impolite)
expressions. The term "lyrics" is not from this letter, as I realized when
re-read it now. However, it is a true Landau expression he applied to
existence theorems. I simply picked it up somewhere else.
Finally I apologize, that my translation is not at all
professional. Have fun. Alexei
This letter has been written to be sent to the rector of one of Moscow
technical universities as a response to its program in mathematics, that
probably had been made public at that time. No more details on the origin
and effect of this text is available*.
Taking into account the importance of mathematics for
physicists, (as it is of general knowledge, physicists experience a need in
a calculating, analytical mathematics), the mathematicians, however, for
incomprehensible for me reason just fob us logical exercises off as an
involuntary purchase**. In your curriculum this statement is directly
expressed in a special note in the beginning of the program. It seems me
that it is high time to teach physicists things that they consider necessary
for themselves, rather than save their souls against their own wish. I do
not want to argue against the scholastic mediaeval idea that one can
allegedly learn to think by the way of learning unnecessary things.
I strongly think that all existence theorems, too rigorous
proofs and so on, and so on... should be completely exorcised from the
mathematics studied by physicists. For this reason I will not specially
focus on those numerous points of your program that are in a drastic
contradiction to this point of view. I will only make few additional notes.
Historical introduction makes a strange impression. It is
self-understood that communicating interesting historical details may only
make the lectures more interesting. It is not clear, however, why this is
considered as a program point. I hope that this is not intended to be
included into tests. The vector analysis is placed between the multiple
integrals. I have nothing against such a combination. I hope, however, that
it makes no damage to a very necessary knowledge of formal formulae of the
vector analysis. The program concerning series is especially overloaded by
unnecessary things, in which the scarce useful information about Fourier
series and integrals sink. The course of the so-called, mathematical
physics*** I would make optional. It is not possible to require
physicists-experimentalists to master such things. One should also say that
the program is too much overloaded. The necessity in the course of the
probability theory is rather doubtful. Physicists anyway teach (their
students, AB) all they need within the courses of statistical physics and
quantum mechanics.
Anyway the presented program is got flooded with uselessness.
For this reason I think that teaching of mathematics needs a most serious
reformation.
*I believe that this letter has been sent to the rector, but made no effect
at all. Landau was not yet that broadly famous at that time. And if he were,
nothing would be changed.
**Involuntary purchase - a practice existed in USSR, when a shop customer
had to buy an unnecessary good, in order to be able to buy a necessary, but
rare one.
*** In the USSR this was the subject unifying partial differential equations
and complex numbers theory.
Alexei BOULBITCH, Dr., habil.
IEE S.A.
ZAE Weiergewan,
11, rue Edmond Reuter,
L-5326 Contern, LUXEMBOURG
Office phone : +352-2454-2566
Office fax: +352-2454-3566
mobile phone: +49 151 52 40 66 44
e-mail: alexei.boulbitch at iee.lu<mailto:alexei.boulbitch at iee.lu>